Lazy Mathematicians?

This post focuses on efficient mathematics, why it is valuable, and the challenges involved in producing it.

“A good mathematician is a lazy mathematician,” at least according to some of my old math professors. It’s a jokey phrase, but one which contains an important element of truth: good mathematicians find ways to do mathematics quickly and easily. In other words, they make their discipline more efficient. But, it turns out that making mathematics efficient is a hard task, requiring mathematicians to develop new and innovative ideas, as well as co-operating to disseminate them.

First, what does efficient, as opposed to inefficient, mathematics look like? We can get an idea by comparing two ways of finding a solution to equations of the form $x^3 + mx =n.$ The first is from Gerolamo Cardano’s¹ book Ars Magna (The Great Art), published in 1545:

Raise a third part of the number of things to a cube, to which you add the square of half the number of the equation, and take the root of all of it, that is, the square root, which you put twice, and to one you add half of the number, which you multiplied by itself, and from the other you subtract the same half, and you will have a binome with its apotome, whence the cube root of the apotome has been subtracted from the cube root of its binome, the difference that remains, that is the solution²

To get a better feel for Cardano’s method, let’s apply it to find a solution to the particular equation $x^3 + 6x =20.$ First, a few preliminaries. When Cardano speaks of “the number of things,” he is referring to the coefficient of “ $x$ ,” and when he speaks of “the number of the equation,” he is referring to the constant term. For example, in $x^3 + 6x =20$ , the “number of things” is 6 and “the number of the equation” is 20.

The first instruction Cardano gives us is to “Raise a third part of the number of things to a cube,” which yields $(6/3)^3 = 2^3 =8.$ Next, to this we “add the square of half the number of the equation,” i.e. we add $(20/2)^2=100$ to 8, giving 108. Then, Cardano tells us that we take the square root of this, and obtain $\sqrt{108}$ . To form the binome, we add on “half of the number, which you multiplied by itself,” by which Cardano means $20/2=10$ . Thus the binome is $\sqrt{108}+10$ . To form the apotome, we instead subtract 10, yielding $\sqrt{108}-10$ . We get the solution to the equation when “the cube root of the apotome has been subtracted from the cube root of its binome.” In other words, the solution³ is $\sqrt[3]{(\sqrt{108}+10)} - \sqrt[3]{(\sqrt{108} -10)}$ .

Now that we’ve seen Cardano’s method, let’s look at the modern version⁴: If we have a cubic of the form $x^3 + mx =n$ , a solution is given by the formula: $x = \sqrt[3]{\sqrt{\left(\frac{m}{3}\right)^{3}+\left(\frac{n}{2}\right)^{2}}+\frac{n}{2}} - \sqrt[3]{\sqrt{\left(\frac{m}{3}\right)^{3}+\left(\frac{n}{2}\right)^{2}}-\frac{n}{2}}.$

If we want to use this to find a solution to the equation from before, $x^3 + 6x =20$ , we just have to substitute “6” for “ $m$ ” and “20” for “ $n$ .” And when we do so, after simplifying, we see we get the same result as before: $\sqrt[3]{(\sqrt{108}+10)} - \sqrt[3]{(\sqrt{108} -10)}.$

The modern way of solving the cubic equation is really just a translation of Cardano’s original method. But, it is much more efficient, requiring far less effort to use. In particular, the modern formula is much easier to parse, taking up just one line compared to Cardano’s paragraph of prose. Moreover, it allows you to see at a glance how and where coefficients from the original cubic $x^3 + mx = n$ get used to construct the solution. For example, we can see immediately that the constant coefficient, $n$ , occurs four times in the expression for the solution. This makes it very easy to apply the formula to solve particular cubics—we are given the form of the solution and then just need to substitute appropriate values for “ $m$ ” and “ $n$ ,” before finally simplifying the resulting expression. With Cardano’s recipe, however, we need to work harder to uncover the same information and must pay close attention to make sure we are interpreting his instructions correctly.

As human beings, the total of amount of effort we can expend is limited, which is why efficiency is so important. If a piece of mathematics is less efficient, it requires more effort for us to use and apply, which means we have less effort to spend elsewhere, such as on adapting or generalizing our work. On the other hand, if a piece of mathematics is more efficient, it saves our energy, allowing us to use it elsewhere, and hopefully bring about the next mathematical break through!

But, a mathematician aiming to streamline the discipline faces tough challenges. To succeed, she first needs to be perceptive, having an initial flash of insight that allows her to tweak existing math to make it more efficient, or to invent completely new and efficient math. However, already there is a complication. If you want to design efficient math, you should know the purpose to which it will be put—something which is difficult to know fully in advance. Indeed, the same piece of mathematics can have unexpected, but very important, applications. For example, our mathematician might develop a piece of mathematics that solves a very particular problem, but later discovers that it can also be used to solve different problems in other areas. If she has designed her mathematics with the first domain in mind, it may provide a very efficient way of tackling that problem. However, its efficiency won’t necessarily carry over to the problems in other domains. Without a way to see into the future, though, there’s little our mathematician can do about this.

Moreover, while our mathematician might be excited about her ideas for efficient mathematics, she must have a great deal of patience. Mathematicians, just like everyone else, make mistakes,⁵ and what initially seems like a slick short-cut can turn out to be problematic. To avoid this, our mathematician should take the time to produce clear, precise rules of use for her new math, and make use of them to check that her work doesn’t have unexpected and unwanted consequences.

Finally, our mathematician must be persuasive! As mathematics is a communal activity, it’s not enough for her to create an efficient piece of mathematics—she has to convince her colleagues to adopt it as well. Otherwise, she is the only one who will benefit from it, and other mathematicians will be wasting effort that could be put to better use. Sometimes our mathematician may find it easy to “sell” her approach to her colleagues—they may look at her work and instantly see the value of her approach. Other times, especially if her work upends an established way of doing things, it can be much harder, as her colleagues may resist changing established practices. And it is reasonable for them to resist—after all, such changes will mean they have to expend effort to learn and understand the new approach when the old one had worked fine. But, if she can show them that they’ll save more effort with the new method than they’ll have to expend to learn it, she should be able to win them over.

So, good mathematicians may be lazy in the sense that they find quicker and easier ways to do mathematics. But, to do this successfully, they need to be insightful, patient, persuasive, and ideally have a way to see into the future! So perhaps a good mathematician isn’t quite so lazy after all!

Notes

[1]. Cardano was an extremely interesting character! You can read more about his life in an article by J J O’Connor and E F Robertson at the MacTutor History of Mathematics site here.

[2] Cardano Ars Magna, translation from pg 9 of Stedall’s From Cardano’s Great Art to Lagrange’s Reflections: Filling a Gap in the History of Algebra, European Mathematics Society, 2011.

[3] $\sqrt[3]{(\sqrt{108}+10)} - \sqrt[3]{(\sqrt{108} -10)}$ further simplifies to 2.

[4] See pg 10 of Stedall’s From Cardano’s Great Art to Lagrange’s Reflections: Filling a Gap in the History of Algebra, European Mathematics Society, 2011.

[5] See this thread on MathOverflow for some examples of interesting mathematical mistakes.